Masteries

Offense:
1

Defense:
25

Utility:
4

Ah Amumu, probably my favorite tank to play. He was the first champion I actually bought using my IP. During my first while playing him I used the sunfire cape build posted by Kryptix. It was fairly solid and is why I still start with Philosopher's Stone -> Boots of Speed. However, after enough games with Amumu I found that one Cape is enough to help you farm and there are plenty of other helpful items that can do more than just aura damage while also providing added boosts.

Skills/Spells:
You can choose to take your stun or Tantrum first, I like taking tantrum since I find that Amumu doesn't have much to offer for a level one fight and you'll end up with Bandage after you level up with help from Tantrum anyways. If you like sitting in front of creep waves, hit the minions evenly then try to get a last hit on 2 or more with a single Tantrum. After getting sunfire cape, your poor dps is remedied, and you can farm like a mad.. sad mummy. Get your ult at every opportunity prioritizing skills in the R->E->W->Q order. Cleanse out of enemy ignites/etc, and Ghost for mobility in teamfight/chasing/escaping. Ignite is why you took cleanse over other spells. You do not have as much armor as Stacked Cape Amumu so reduced healing will eat you. :p

Items:
Stone, Boots, and Cape are your core 3 items, you can choose the other items in order of what you need most, although I find the order I put up is usually how I end up taking the items. (Note: You can take counter boots if the team is really caster/melee heavy.)
Try to hold off on Force of Nature until you get at least one more of the Giant's Belt items. After getting a Mana Manipulator and evntually Soul Shroud, you and your lane partner (you didn't take solo right?) should have little worries with mana. Rylai's will make your Despair a perma-slow while Shroud's Aura is really helpful. After Sunfire, your team will probably be in the ganking/mini fight phase and both Rylai's/Shroud will help in these fights. If the game is still going on, sell the stone and buy the super green armor! Warmog's will add flat health regen AND more health for your Force of Nature to regen. I've won several 2v1 fights thanks to the build.

Runes/Masteries:
I found dodge runes with Nimbleness and Swiftness boots helps with sticking on to single targets and with running from a bad situation. If you find you really like dodge, you can take the Quints as well and swap your Swiftness Boots for Tabis. You can choose more defensive options for your Glyphs and Marks but with scaling AP glyphs, Rylai's and Champion level 13, your Despair does an additional 0.5% hp per tick. Since your main damage output is magic(including Cape damage) I find magic pen helps the most. Force of Nature should be enough magic resist to not have to take it in your runes and the defense provided by Marks/Glyphs is pretty shabby.

Thanks for checking out the build, try it out if you're an Amumu player and send some criticism back if you did. :)